A note on Newton's problem of minimal resistance for convex bodies

Abstract

We consider the following problem: minimize the functional $\int_\Omega f(\nabla u(x))\, dx$ in the class of concave functions $u: \Omega \to [0,M]$, where $\Omega \subset \mathbb{R}^2$ is a convex body and $M > 0$. If $f(x) = 1/(1 + |x|^2)$ and $\Omega$ is a circle, the problem is called Newton's problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of $\partial\Omega$ are regular and ${|x|f(x)}/(|y|f(y)) \to +\infty$ as $|x|/|y| \to 0$ then a solution $u$ to the problem satisfies $u\rfloor_{\partial\Omega} = 0$. This result proves the conjecture stated in 1993 for Newton's problem.